The Quaternions and their Applications Rob Eimerl April 28, 2015 Abstract This paper is an attempt to summarize Quaternion Algebras. The first part looks at their origins and certain properties of these algebras are examined, from the roots of Polynomials over the Quaternions to how to construct a quaternion algebra and Frobenius' theorem. The second part of this paper looks at applications of quaternion algebras. For the sake of brevity proofs are omitted from this paper but are cited License: CC-BY-NC CONTENTS CONTENTS Contents 1 Introduction 3 2 Quaternion Algebra 3 2.1 Preliminaries . 3 2.2 The Real Quaternions (H)........................... 4 2.3 Generalized Quaternion Algebra (Q) . 4 2.4 Polynomials in H . 5 2.5 Extensions of Quaternion Algebras . 6 3 Applications of Quaternion Algebra 7 4 Conclusion 8 License: CC-BY-NC 2 QUATERNION ALGEBRA 1 Introduction In late 1843 the great Irish mathematician William Hamilton was struggling over a prob- lem that had confounded him for nearly a decade, how to multiply triplets. In 1835 he had developed techniques that allowed complex numbers to be treated as pairs of real numbers (a + bia; b 2). Intrigued by the relation between complex numbers (C) and 2-dimensional geometry, Hamilton tried for years to develop an analogous 3-dimensional algebra. To appreciate the problem consider a vector space (V ), with the basis a; b; c, and add the additional operation * (vector multiplication that distributes over addition). If we take v = a + b + c (the simplest linear combination of every basis vector) by closure v2 should also be an element of V . However, when we compute v2 we find v2 = v(v) = (a + b + c) ∗ (a + b + c) = a(a + b + c) + b(a + b + c) + c(a + b + c) (1) = a2 + ab + ac + ba + b2 + bc + ca + cb + c2 = (a2 + b2 + c2) + ab + ac + ba + bc + ca + cb This is a hopelessly confusing result, that doesn't make sense until multiplication of the basis vectors is defined. But how should such an operation be defined? We need six products that preserve closure and the other properties of V . On the 16th of October, Hamilton had a brain-wave. While walking to a meeting of the Royal Irish Academy, he felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i; j; k exactly as I have used them ever since. He famously carved these equations into the stone of Brougham Bridge in Dublin i2 = j2 = k2 = ikj = −1; these first relations quickly led to secondary ones: ij = −ji = k; ik = −ki = j; jk = −kj = i: Hamilton called 1; i; j; k the quaternions and with them built the worlds first non-commutative algebra1, opening the door to a wide range of new algebraic structures and ideas, and in a very real way to development of Abstract Algebra as field of study[4]. 2 Quaternion Algebra 2.1 Preliminaries Algebra: Let F be a field, and let A be a vector space over F with an additional binary operation (∗) from A × A to A. Then A is an algebra over F if the following expressions hold for any three elements x; y; z 2 A, and for a, b 2F: [8] License: CC-BY-NC 2.2 The Real Quaternions (H) 2 QUATERNION ALGEBRA 1. Right distributivity: (x + y) ∗ z = x ∗ z + y ∗ z 2. Left distributivity: x ∗ (y + z) = x ∗ y + x ∗ z 3. Compatibility with scalars: (ax) ∗ (by) = (ab)(x ∗ y). 2.2 The Real Quaternions (H) The first algebraic structure Hamilton built was the Real Quaternions, H: H = fq = a + bi + cj + dk : a; b; c; d 2 R; f1; i; j; kg 2 Qg H is a 4-dimensional vector space over R with the standard forms of addition and scalar multiplication, that is then boosted into a ring by defining quaternion multiplication (indicated by juxtaposition) via i2 = j2 = 1; ij = ji = k[4], such that q1q2 = (a1a2 − b1b2 − c1c2 − d1d2) + (a b − a b − c d − d c )i 1 2 2 1 1 2 1 2 (2) + (a1c2 − b1d2 + c1a2 − d1b2)j + (a1d2 + b1c2 − c1b2 + d1a2)k 2.3 Generalized Quaternion Algebra (Q) This leads to a generalized Quaternion Algebra (Q) by replacing with any field F, and α,β redefining basis multiplication. A quaternion algebra over a field F is written as Q = F and elements of Q have the form q = a + bi + cj + dk Quaternion multiplication is defined by α and β (nonzero elements of F , a = b not excluded) such that the new multiplication rules are i2 = a; j2 = bij = −ji = k At this point it is worth defining certain quantities associated with quaternion algebras a;b [4] Let q 2 F ; q = a + bi + cj + dk Quaternion Conjugate: q¯ = a + (−b)i + (−c)j + (−d)k License: CC-BY-NC 2.4 Polynomials in H 2 QUATERNION ALGEBRA Complex Conjugate: q∗ = a∗ + (−b∗)i + (−c∗)j + (−d∗)k a;b Norm: N : F ! F such that N(q) =qq ¯ = qq¯ = a2 + (−αb2) + (βc2) + αβd2 Norm form: Coefficients of N(q) expressed < 1; α; β; αβ > −1 1 It should be clear that whenever N(q) 6= 0 the element q is invertible, q = N(q) q¯. Indeed q is invertible if and only if N(q) 6= 0. This leads us to our first theorem of Quaternion Algebras. a;b Theorem 1: The quaternion algebra F is a division algebra if and only if N(q) = 0 implies q = 0 [4]. The Norm form also gives a way to test whether two quaternion algebras are isomor- phic. Theorem 2: Quaternion Algebras are isomorphic as algebras if and only if their norm forms are isometric. Two forms, Q1 : V1 ! F and Q2 : V2 ! F are isometric if their exists a vector space isomorphism φ : V1 ! V2, such that Q2(φ(x)) = Q1(x) for all x 2 V1. [4] Example: a; b b; a ∼= ; F F their norm forms < 1; a; b; ab > and < 1; b; a; ab > are isometric. 2.4 Polynomials in H The subject of algebra began as attempts to solve various equations,particularly polyno- mial equations. The Fundamental Theorem of Algebra states: Any polynomial of degree n with coefficients in any field F can have at most n roots in F. For polynomials with coefficients from H the situation is somewhat different. Due to the lack of commutativity polynomials become commensurately more complicated, in just degree two we may have terms like ax2; xax; x2a; axbx. However, there is a Fundamental Theorem of Algebra for H which says that if the polynomial has only one term of highest degree then there exists a root in H[7]. In the context of solving polynomial equations we should mention the Wedderburn Factorization Theorem for polynomials over division algebras. Wedderburn Factorization Theorem[4] Let D be a division ring with center F and let p(x) be an irreducible monic polynomial of degree n with coefficients from the field F. If there exists d 2 D such that p(d) = 0 then we can write p(x) = (td1)(td2)(td3)(tdn) and each di is conjugate to d1; there exist License: CC-BY-NC 2.5 Extensions of Quaternion Algebras 2 QUATERNION ALGEBRA 1 nonzero elements si 2 D, such that di = sidsi for each i. This theorem says that if the polynomial has one root in D then it factorizes completely as a product of linear factors over D. This is a major improvement compared to the situation in field theory where only one root of a polynomial is guaranteed in a given field. 2.5 Extensions of Quaternion Algebras In December 1843 John Graves extended the quaternions and discovered the octonions O, an 8-dimensional algebra over R which is both noncommutative and nonassociative[2]. The process of going from R to C, from C to H, and from H to O, known as Cayley- Dickson Doubling, is a way of extending an algebra A to a new algebra, KD(A), and preserving all operations (addition, scalar multiplication, element multiplication and the norm), such that A is a subalgebra of KD(A). If A has a unity element Θ then so does KD(A) and the extension can be expressed; KD(A; Θ) = A ⊕ Aµ where µ is a root of unity. Each extension is a 2 degree extension of the preceding algebra.[1] Example: Extensions of R KD(R) = R + Ri = a + bi = C (3) KD(C) = C + Cj = (a + bi) + (c + di)j = Q Generalizing this example to any field F we can define the Quaternion algebra over F as KD(KD(F; Θ1); Θ2), where Θi 2 F n f0g Note that at each stage of the doubling process one of the intrinsic properties of the base algebra is lost. From R to C we lose ordering, from C to H commutativity and from H to O associativity. If we apply the CayleyDickson doubling process to the octonions we obtain a structure called the sedenions, which is a 16-dimensional nonassociative algebra, However the sedenions are not a division ring and not very conducive for working with. The mathematician Frobenius took this idea of subalgebras and found an interesting resul about Real Division Algebras.[3] Theorem (Frobenius): Suppose A is an algebra with unit over the field R of reals. Assume that the algebra A is without divisors of zero. If each element x 2 A is algebraic with respect to the field R then the algebra is isomorphic with one of the classical division algebras R; C; or Q.